projections and closed subspaces
Given a topological complement of , there exists a unique continuous projection onto such that for all and .
The projection in the second part of the above theorem is sometimes called the projection onto along .
The above result can be further improved for Hilbert spaces.
Theorem 2 - Let be a Hilbert space and a closed subspace. Then, is topologically complemented in if and only if there exists an orthogonal projection onto (which is unique).
Corollary - Let be a Hilbert space and a closed subspace. Then, there exists a unique orthogonal projection onto . This establishes a bijective correspondence between orthogonal projections and closed subspaces.
|Title||projections and closed subspaces|
|Date of creation||2013-03-22 17:52:57|
|Last modified on||2013-03-22 17:52:57|
|Last modified by||asteroid (17536)|
|Synonym||projection along a closed subspace|
|Synonym||orthogonal projections onto Hilbert subspaces|